Abstract
We show, by means of numerical and analytical methods, that media with a repulsive nonlinearity which grows from the center to the periphery support a remarkable variety of previously unknown complex stationary and dynamical three-dimensional (3D) solitary-wave states. Peanut-shaped modulation profiles give rise to vertically symmetric and antisymmetric vortex states, and novel stationary hybrid states, built of top and bottom vortices with opposite topological charges, as well as robust dynamical hybrids, which feature stable precession of a vortex on top of a zero-vorticity soliton. The analysis reveals stability regions for symmetric, antisymmetric, and hybrid states. In addition, bead-shaped modulation profiles give rise to the first example of exact analytical solutions for stable 3D vortex solitons. The predicted states may be realized in media with a controllable cubic nonlinearity, such as Bose–Einstein condensates.
Highlights
Self-trapping of three-dimensional (3D) confined modes in optics [1,2,3], Bose–Einstein condensates (BECs) [4,5,6], ferromagnetic media [7], superconductors [8], semiconductors [9], baryonic matter [10], and general field theory [11, 12] is a fundamental problem of nonlinear physics
An apparent condition is that an attractive, or self-focusing, nonlinearity is necessary for the creation of localized states; the attractive cubic nonlinearity simultaneously gives rise to collapse [13] of localized modes in higher-dimensional settings and, to strong azimuthal modulational instability of states with intrinsic vorticity [14], making the search for stable 3D fundamental and topological solitons in materials with the cubic (Kerr) nonlinearity a challenging issue
It is relevant to mention a very recent result concerning 2D localized modes created by the self-focusing cubic nonlinearity in the free space: while a common belief was that they might never be stable, it has been demonstrated in [20] that mixed vortex-fundamental modes in a system of two coupled Gross–Pitaevskii equations modeling the spin-orbit-coupled BEC can be stable in the 2D free space
Summary
Self-trapping of three-dimensional (3D) confined modes (solitons or, more properly, solitary waves) in optics [1,2,3], Bose–Einstein condensates (BECs) [4,5,6], ferromagnetic media [7], superconductors [8], semiconductors [9], baryonic matter [10], and general field theory [11, 12] is a fundamental problem of nonlinear physics. Our analysis reveals that 3D media with a repulsive nonlinearity growing from two symmetric minima to the periphery make it possible to create complex but, stable static and dynamical self-trapped topological modes, in the form of fundamental and vortical dipoles, stationary vortex–antivortex hybrids, and precessing hybrids built as a vortex sitting on top of a zero-vorticity mode. These are remarkable, novel species of 3D localized modes, which have not been reported before in any other systems. The modulation profile (3) is adopted here as it makes it possible to obtain families of stationary vortex modes in an almost exact analytical form, by means of the TFA, supporting numerical findings
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